The behavior of polyatomic molecules around their equilibrium positions can be regarded as that of quantum-coupled anharmonic oscillators. Solving the corresponding Schrödinger equations enables the interpretation or prediction of the experimental spectra of molecules. In this study, we developed a novel approach for solving the excited states of anharmonic vibrational systems. The normal coordinates of the molecules are transformed into new coordinates through a normalizing flow parameterized by a neural network. This facilitates the construction of a set of orthogonal many-body variational wavefunctions. This methodology has been validated on an exactly solvable 64-dimensional coupled harmonic oscillator, yielding numerical results with a relative error of 10−6. The neural canonical transformations are also applied to calculate the energy levels of two specific molecules, acetonitrile (CH3CN) and ethylene oxide (C2H4O). These molecules involve 12 and 15 vibrational modes, respectively. A key advantage of this approach is its flexibility concerning the potential energy surface, as it requires no specific form. Furthermore, this method can be readily implemented on large-scale distributed computing platforms, making it easy to extend to investigating complex vibrational structures.
REFERENCES
1.
J. M.
Bowman
, “Self-consistent field energies and wavefunctions for coupled oscillators
,” J. Chem. Phys.
68
, 608
–610
(1978
).2.
J. M.
Bowman
, “The self-consistent-field approach to polyatomic vibrations
,” Acc. Chem. Res.
19
, 202
–208
(1986
).3.
R.
Gerber
and M.
Ratner
, “A semiclassical self-consistent field (SC SCF) approximation for eigenvalues of coupled-vibration systems
,” Chem. Phys. Lett.
68
, 195
–198
(1979
).4.
T. C.
Thompson
and D. G.
Truhlar
, “Optimization of vibrational coordinates, with an application to the water molecule
,” J. Chem. Phys.
77
, 3031
–3035
(1982
).5.
K. M.
Christoffel
and J. M.
Bowman
, “Investigations of self-consistent field, SCF CI and virtual stateconfiguration interaction vibrational energies for a model three-mode system
,” Chem. Phys. Lett.
85
, 220
–224
(1982
).6.
G.
Rauhut
, “Configuration selection as a route towards efficient vibrational configuration interaction calculations
,” J. Chem. Phys.
127
, 184109
(2007
).7.
Y.
Scribano
and D. M.
Benoit
, “Iterative active-space selection for vibrational configuration interaction calculations using a reduced-coupling VSCF basis
,” Chem. Phys. Lett.
458
, 384
–387
(2008
).8.
N.
Gohaud
, D.
Begue
, C.
Darrigan
, and C.
Pouchan
, “New parallel software (P_Anhar) for anharmonic vibrational calculations: Application to (CH3Li)2
,” J. Comput. Chem.
26
, 743
–754
(2005
).9.
D.
Begue
, N.
Gohaud
, R.
Brown
, and C.
Pouchan
, “An single program multiple data strategy for calculation of anharmonic vibrations
,” J. Math. Chem.
40
, 197
–211
(2006
).10.
D.
Bégué
, N.
Gohaud
, C.
Pouchan
, P.
Cassam-Chenaï
, and J.
Liévin
, “A comparison of two methods for selecting vibrational configuration interaction spaces on a heptatomic system: Ethylene oxide
,” J. Chem. Phys.
127
, 164115
(2007
).11.
P.
Cassam-Chenaï
and J.
Liévin
, “Alternative perturbation method for the molecular vibration–rotation problem
,” Int. J. Quantum Chem.
93
, 245
–264
(2003
).12.
P.
Cassam-Chenaï
, “Ab initio predictions for the Q-branch of the methane vibrational ground state
,” J. Quant. Spectrosc. Radiat. Transfer
82
, 251
–277
(2003
).13.
S.
Manzhos
, K.
Yamashita
, and T.
Carrington
, Jr., “Using a neural network based method to solve the vibrational Schrödinger equation for H2O
,” Chem. Phys. Lett.
474
, 217
–221
(2009
).14.
K.
Yagi
, M.
Keçeli
, and S.
Hirata
, “Optimized coordinates for anharmonic vibrational structure theories
,” J. Chem. Phys.
137
, 204118
(2012
).15.
K.
Ishii
, T.
Shimazaki
, M.
Tachikawa
, and Y.
Kita
, “Development of anharmonic vibrational structure theory using backflow transformation
,” Chem. Phys. Lett.
787
, 139263
(2022
).16.
A.
Leclerc
and T.
Carrington
, “Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices
,” J. Chem. Phys.
140
, 174111
(2014
).17.
P. S.
Thomas
and T.
Carrington
, Jr., “Using nested contractions and a hierarchical tensor format to compute vibrational spectra of molecules with seven atoms
,” J. Phys. Chem. A
119
, 13074
–13091
(2015
).18.
M.
Rakhuba
and I.
Oseledets
, “Calculating vibrational spectra of molecules using tensor train decomposition
,” J. Chem. Phys.
145
, 124101
(2016
).19.
A. V.
Knyazev
, “Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method
,” SIAM J. Sci. Comput.
23
, 517
–541
(2001
).20.
M.
Odunlami
, V.
Le Bris
, D.
Bégué
, I.
Baraille
, and O.
Coulaud
, “A-VCI: A flexible method to efficiently compute vibrational spectra
,” J. Chem. Phys.
146
, 214108
(2017
).21.
Y.
Saleh
, Á. F.
Corral
, A.
Iske
, J.
Küpper
, and A.
Yachmenev
, “Computing excited states of molecules using normalizing flows
,” arXiv:2308.16468 (2023
).22.
J. M.
Bowman
, T.
Carrington
, and H.-D.
Meyer
, “Variational quantum approaches for computing vibrational energies of polyatomic molecules
,” Mol. Phys.
106
, 2145
–2182
(2008
).23.
A.
Zoccante
, P.
Seidler
, M. B.
Hansen
, and O.
Christiansen
, “Approximate inclusion of four-mode couplings in vibrational coupled-cluster theory
,” J. Chem. Phys.
136
, 204118
(2012
).24.
I. H.
Godtliebsen
, B.
Thomsen
, and O.
Christiansen
, “Tensor decomposition and vibrational coupled cluster theory
,” J. Phys. Chem. A
117
, 7267
–7279
(2013
).25.
N. K.
Madsen
, I. H.
Godtliebsen
, S. A.
Losilla
, and O.
Christiansen
, “Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations
,” J. Chem. Phys.
148
, 024103
(2018
).26.
N. K.
Madsen
, R. B.
Jensen
, and O.
Christiansen
, “Calculating vibrational excitation energies using tensor-decomposed vibrational coupled-cluster response theory
,” J. Chem. Phys.
154
, 054113
(2021
).27.
A.
Erba
, J.
Maul
, M.
Ferrabone
, R.
Dovesi
, M.
Rérat
, and P.
Carbonnière
, “Anharmonic vibrational states of solids from DFT calculations. Part II: Implementation of the VSCF and VCI methods
,” J. Chem. Theory Comput.
15
, 3766
–3777
(2019
).28.
N. M.
Witriol
, “Canonical transformations and molecular structure calculations
,” Int. J. Quantum Chem.
6
, 145
–152
(1972
).29.
H.
Xie
, L. Z.
null
, and L.
Wang
, “Ab-initio study of interacting fermions at finite temperature with neural canonical transformation
,” J. Mach. Learn.
1
, 38
–59
(2022
).30.
H.
Xie
, L.
Zhang
, and L.
Wang
, “m* of two-dimensional electron gas: A neural canonical transformation study
,” SciPost Phys.
14
, 154
(2023
).31.
H.
Xie
, Z.-H.
Li
, H.
Wang
, L.
Zhang
, and L.
Wang
, “Deep variational free energy approach to dense hydrogen
,” Phys. Rev. Lett.
131
, 126501
(2023
).32.
L.
Dinh
, D.
Krueger
, and Y.
Bengio
, “NICE: Non-linear independent components estimation
,” arXiv:1410.8516 (2014
).33.
M.
Germain
, K.
Gregor
, I.
Murray
, and H.
Larochelle
, “Made: Masked autoencoder for distribution estimation
,” in Proceedings of the 32nd International Conference on Machine Learning, Proceedings of the Machine Learning Research
(Proceedings of Machine Learning Research
, 2015
), Vol. 37
, pp. 881
–889
.34.
D.
Rezende
and S.
Mohamed
, “Variational inference with normalizing flows
,” in Proceedings of the 32nd International Conference on Machine Learning, Proceedings of the Machine Learning Research
(Proceedings of Machine Learning Research
, 2015
), Vol. 37
, pp. 1530
–1538
.35.
L.
Dinh
, J.
Sohl-Dickstein
, and S.
Bengio
, “Density estimation using real NVP
,” arXiv:1605.08803 (2016
).36.
G.
Papamakarios
, “Neural density estimation and likelihood-free inference
,” arXiv:1910.13233 (2019
).37.
G.
Papamakarios
, E.
Nalisnick
, D. J.
Rezende
, S.
Mohamed
, and B.
Lakshminarayanan
, “Normalizing flows for probabilistic modeling and inference
,” J. Mach. Learn. Res.
22
, 1
–64
(2021
).38.
W.
Lei
, Generative Models for Physicists, 2018
.39.
D.
Pfau
, S.
Axelrod
, H.
Sutterud
, I.
von Glehn
, and J. S.
Spencer
, “Natural quantum Monte Carlo computation of excited states
,” arXiv:2308.16848 [physics.comp-ph] (2024
).40.
K.
Cranmer
, S.
Golkar
, and D.
Pappadopulo
, “Inferring the quantum density matrix with machine learning
,” arXiv:1904.05903 [quant-ph] (2019
).41.
F.
Becca
and S.
Sorella
, Quantum Monte Carlo Approaches for Correlated Systems
(Cambridge University Press
, 2017
).42.
G.
Herzberg
, Molecular Spectra and Molecular Structure
(V.D. Nostrand Company Inc.
, New York
, 1960
), Vol. 2
.43.
I.
Nakagawa
and T.
Shimanouchi
, “Rotation–vibration spectra and rotational, Coriolis coupling, centrifugal distortion and potential constants of methyl cyanide
,” Spectrochim. Acta
18
, 513
–539
(1962
).44.
M.
Koivusaari
, V.-M.
Horneman
, and R.
Anttila
, “High-resolution study of the infrared band ν8 of CH3CN
,” J. Mol. Spectrosc.
152
, 377
–388
(1992
).45.
A.
Tolonen
, M.
Koivusaari
, R.
Paso
, J.
Schroderus
, S.
Alanko
, and R.
Anttila
, “The infrared spectrum of methyl cyanide between 850 and 1150 cm−1: Analysis of the ν4, ν7, and 3ν18 bands with resonances
,” J. Mol. Spectrosc.
160
, 554
–565
(1993
).46.
S. S.
Andrews
and S. G.
Boxer
, “Vibrational Stark effects of nitriles I. Methods and experimental results
,” J. Phys. Chem. A
104
, 11853
–11863
(2000
).47.
R.
Paso
, R.
Anttila
, and M.
Koivusaari
, “The infrared spectrum of methyl cyanide between 1240 and 1650 cm−1: The coupled band system ν3, ν, and ν7 + ν
,” J. Mol. Spectrosc.
165
, 470
–480
(1994
).48.
J.
Duncan
, D.
McKean
, F.
Tullini
, G.
Nivellini
, and J.
Perez Peña
, “Methyl cianide: Spectroscopic studies of isotopically substituted species, and the harmonic potential function
,” J. Mol. Spectrosc.
69
, 123
–140
(1978
).49.
J.
Nakagawa
, K. M.
Yamada
, and G.
Winnewisser
, “Infrared diode-laser spectrum of the CH3CN ν2 band. Analysis of local resonances with + 2, ν4 + + , ν4 + , ν4 + , and 2 + states
,” J. Mol. Spectrosc.
112
, 127
–141
(1985
).50.
D.
Begue
, P.
Carbonniere
, and C.
Pouchan
, “Calculations of vibrational energy levels by using a hybrid ab initio and DFT quartic force field: Application to acetonitrile
,” J. Phys. Chem. A
109
, 4611
–4616
(2005
).51.
G.
Avila
and T.
Carrington
, “Using nonproduct quadrature grids to solve the vibrational Schrödinger equation in 12D
,” J. Chem. Phys.
134
, 054126
(2011
).52.
A.
Komornicki
, F.
Pauzat
, and Y.
Ellinger
, “Ab initio prediction of structures, force constants and vibrational frequencies. Saturated three-membered rings cyclopropane, ethylene oxide, and ethyleneimine
,” J. Phys. Chem.
87
, 3847
–3857
(1983
).53.
M. A.
Lowe
, J. S.
Alper
, R.
Kawiecki
, and P.
Stephens
, “Scaled ab initio force fields for ethylene oxide and propylene oxide
,” J. Phys. Chem.
90
, 41
–50
(1986
).54.
S.
Rebsdat
and D.
Mayer
, Ethylene Oxide
(Ullmann’s Encyclopedia of Industrial Chemistry
, 2000
).55.
N.
Cant
and W.
Armstead
, “Vibrational assignments and force-field calculations for ethylene oxide
,” Spectrochim. Acta, Part A
31
, 839
–853
(1975
).56.
R. C.
Lord
and B.
Nolin
, “Vibrational spectra of ethylene oxide and ethylene oxide-d4
,” J. Chem. Phys.
24
, 656
–658
(1956
).57.
A.
Schriver
, J.
Coanga
, L.
Schriver-Mazzuoli
, and P.
Ehrenfreund
, “Vibrational spectra and UV photochemistry of (CH2)2O thin films and (CH2)2O in amorphous water ice
,” Chem. Phys.
303
, 13
–25
(2004
).58.
T.
Nakanaga
, “Coriolis interactions and assignments of the fundamental bands in ethylene oxide and ethylene oxide-d4
,” J. Chem. Phys.
73
, 5451
–5458
(1980
).59.
N.
Yoshimizu
, C.
Hirose
, and S.
Maeda
, “Microwave spectrum of ethylene oxide in excited vibrational states
,” Bull. Chem. Soc. Jpn.
48
, 3529
–3532
(1975
).60.
P.
Carbonnière
, A.
Dargelos
, and C.
Pouchan
, “The VCI-P code: An iterative variation–perturbation scheme for efficient computations of anharmonic vibrational levels and IR intensities of polyatomic molecules
,” Theor. Chem. Acc.
125
, 543
–554
(2010
).61.
J.
Bradbury
, R.
Frostig
, P.
Hawkins
, M. J.
Johnson
, C.
Leary
, D.
Maclaurin
, G.
Necula
, A.
Paszke
, J.
VanderPlas
, S.
Wanderman-Milne
, and Q.
Zhang
, JAX: Composable transformations of Python + NumPy programs
, 2018
.62.
The source codes are accessible at https://github.com/zhangqi94/VibrationalSystem and the data are accessible at https://github.com/zhangqi94/VibrationalSystemData.
63.
H. H.
Nielsen
, “The vibration–rotation energies of molecules
,” Rev. Mod. Phys.
23
, 90
–136
(1951
).64.
J. K.
Watson
, “Simplification of the molecular vibration–rotation Hamiltonian
,” Mol. Phys.
100
, 47
–54
(2002
).65.
B. S.
DeWitt
, “Point transformations in quantum mechanics
,” Phys. Rev.
85
, 653
–661
(1952
).66.
M.
Eger
and E.
Gross
, “Point transformations and the many body problem
,” Ann. Phys.
24
, 63
–88
(1963
).67.
F.
Eger
and E.
Gross
, “Point transformations and scattering by a hard core
,” Il Nuovo Cimento
34
, 1225
–1241
(1964
).68.
F.
Lu
, L.
Cheng
, R. J.
DiRisio
, J. M.
Finney
, M. A.
Boyer
, P.
Moonkaen
, J.
Sun
, S. J.
Lee
, J. E.
Deustua
, T. F.
Miller
III, and A. B.
McCoy
, “Fast near ab initio potential energy surfaces using machine learning
,” J. Phys. Chem. A
126
, 4013
–4024
(2022
).69.
A. M.
Alvertis
, J. B.
Haber
, E. A.
Engel
, S.
Sharifzadeh
, and J. B.
Neaton
, “Phonon-induced localization of excitons in molecular crystals from first principles
,” Phys. Rev. Lett.
130
, 086401
(2023
).70.
D. P.
Kingma
and J.
Ba
, “Adam: A method for stochastic optimization
,” in 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, edited by Y. Bengio and Y. LeCun, 2015.71.
S.-H.
Li
, C.-X.
Dong
, L.
Zhang
, and L.
Wang
, “Neural canonical transformation with symplectic flows
,” Phys. Rev. X
10
, 021020
(2020
).72.
E. K. U.
Gross
, L. N.
Oliveira
, and W.
Kohn
, “Rayleigh-Ritz variational principle for ensembles of fractionally occupied states
,” Phys. Rev. A
37
, 2805
–2808
(1988
).73.
C.
Durkan
, A.
Bekasov
, I.
Murray
, and G.
Papamakarios
, “Neural spline flows
,” in Advances in Neural Information Processing Systems 32 (NeurIPS 2019), 33rd Conference on Neural Information Processing Systems, NeurIPS 2019, 08 December 2019--14 December 2019
(Neural Information Processing Systems Foundation, Inc.
, 2019), Vol. 32
, pp. 7511
–7522
.74.
D.
Wu
, L.
Wang
, and P.
Zhang
, “Solving statistical mechanics using variational autoregressive networks
,” Phys. Rev. Lett.
122
, 080602
(2019
).75.
J.-G.
Liu
, L.
Mao
, P.
Zhang
, and L.
Wang
, “Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits
,” Mach. Learn.: Sci. Technol.
2
, 025011
(2021
).76.
P.
Goyal
, P.
Dollár
, R.
Girshick
, P.
Noordhuis
, L.
Wesolowski
, A.
Kyrola
, A.
Tulloch
, Y.
Jia
, and K.
He
, “Accurate, large minibatch SGD: Training ImageNet in 1 hour
,” arXiv:1706.02677 (2017
).77.
B.
Monserrat
, N. D.
Drummond
, and R. J.
Needs
, “Anharmonic vibrational properties in periodic systems: Energy, electron–phonon coupling, and stress
,” Phys. Rev. B
87
, 144302
(2013
).78.
L.
Monacelli
, R.
Bianco
, M.
Cherubini
, M.
Calandra
, I.
Errea
, and F.
Mauri
, “The stochastic self-consistent harmonic approximation: Calculating vibrational properties of materials with full quantum and anharmonic effects
,” J. Phys.: Condens. Matter
33
, 363001
(2021
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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